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A Note on the Theory of Low‐Frequency Waves in a Rotating Stratified Channel

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It is found that the possible low‐frequency, quasigeostrophic motions in a rotating, stratified channel with a wave‐maker at one end include: (i) “standing waves” whose amplitudes are damped exponentially away from the forcing, and (ii) baroclinic internal Kelvin waves, trapped to the right‐hand wall when facing in the direction of phase propagation. The Kelvin waves are excited only if the wave‐maker transfers mean energy to the fluid. The standing waves, on the other hand, carry no energy and thus serve mainly to provide continuity between the wave‐maker and the fluid. When the bottom of the channel is inclined to the horizontal by a small angle , topographic oscillations are possible. These waves behave like topographic Rossby waves if the forcing frequency is greater than sN and if the ratio HN/fL is small, where s=tan,  is the angle of the bottom slope, L is the width of the channel, H is the mean depth, f is the Coriolis parameter, and N is the Brunt‐Vaisala (or buoyancy) frequency. It is determined that topographic Rossby waves cannot exist in the channel if HN/fL0.65. If the wave‐maker frequency is smaller than sN, and if HN/fL~1, the topographic oscillations become bottom‐trapped, decaying away from the bottom boundary in a distance ~λf/N, where λ is the horizontal wavelength. The phase and energy of the bottom‐trapped wave both move to the left of an observer who is facing shallow water. The Kelvin waves are basically unchanged when the bottom is inclined if their down‐channel wavelength is large relative to the width of the channel. The standing oscillations of the flat‐bottom case exist as complex‐horizontal‐wave‐number solutions to the topographic wave dispersion relation. Although these waves have propagating phase when s≠0, they are still trapped to the forcing, and do not transfer net energy from the wave‐maker to the fluid. All three modes are required to solve the general matching conditions for an arbitrary wave‐maker when the channel has a sloping bottom.
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Document Type: Research Article

Affiliations: Massachusetts Institute of Technology Uppsala University

Publication date: December 1, 1976

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